TRUE TRANSLATION FROM PERSIAN TEXT


IJAHP Article: Alitaneh /Theories on coefficient of variation scales triangle and 

normalization of different variables: a new model in development of multiple criteria decision 

analysis 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

283 Vol. 11 Issue 2 2019 

ISSN 1936-6744 
https://doi.org/10.13033/ijahp.v11i2.565 

 

THEORIES ON COEFFICIENT OF VARIATION SCALES 

TRIANGLE AND NORMALIZATION OF DIFFERENT 

VARIABLES: A NEW MODEL IN DEVELOPMENT OF 

MULTIPLE CRITERIA DECISION ANALYSIS 

 

Saeed Alitaneh
1
 

M.Sc., Birjand University, Iran 

Saeed.alitaneh@gmail.com 

 

 

ABSTRACT 

 

This paper is an attempt to solve various problems by the two factors of mean and 

standard deviation (SD) of variables, introducing coefficient of variation (CV) of data 

as the best option for prioritization, scaling, pairwise comparison and normalization 

of quantitative and qualitative variables. An algorithm was built based on a 

coefficient of variation scales triangle (CVST) consisting of natural numbers with 

coefficients of binomial expansion for each line, followed by new and independent 

grading and scaling. In view of the existing factors, the theory provides higher 

generalization and maximum reliability rates in comparison to other methods for 

multiple-criteria decision analysis (MCDA). On the other hand, in the normalization 

process of different variables (i.e. de-scalarization), a precise and infinite model was 

presented based on coefficient of variation scale triangle (multipurpose triangle), in 

such a way that decision makers could work with the software in a more convenient 

and precise manner. Therefore, the proposed theories may be considered as a new 

approach and definition in the valuation and progress of MCDA. 

 

Keywords: theories; coefficient of variation scales triangle (CVST); multiple-criteria 

decision analysis (MCDA) 

 

 

1. Introduction 

The Analytic Hierarchy Process (AHP) is a multi-criteria decision making (MCDM) 

method that helps the decision-maker who is facing a complex problem with multiple 

conflicting and subjective criteria. Several papers have compiled AHP success stories 

in many different fields (Vargas, 1990; Ho, 2008; Golden, Wasil, & Harker, 1989; 

Harker & Vargas, 1990; Shim, 1989; Saaty, & Forman, 1992; Forman, & Gass, 2001; 

Kumar, & Vaidya, 2006; Omkarprasad & Sushil, 2006; Liberatore, & Nydick, 2008; 

Zahedi, 1986). The oldest reference found is Saaty (1972a). After this, Saaty (1977b) 

precisely described the AHP method. The vast majority of applications still use the 

AHP as described in this first publication and are unaware of successive 

developments. The Analytical Hierarchy Process (AHP) technique, developed by 

Thomas L. Saaty, is an MCDA method that helps decision makers make the best 

decisions in the face of complex problems consisting of multiple conflicts and 

internal criteria. The method has already been tested and established in many various 

fields of work (Saaty, 1980c). Recently, the application of AHP in animal science for 

                                                 
1
 The author is deeply indebted to the late Dr. Saaty whose contribution to this field will be 

remembered for years. May his name and memory stay alive throughout eternity. 

 



IJAHP Article: Alitaneh /Theories on coefficient of variation scales triangle and 

normalization of different variables: a new model in development of multiple criteria decision 

analysis 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

284 Vol. 11 Issue 2 2019 

ISSN 1936-6744 
https://doi.org/10.13033/ijahp.v11i2.565 

 

the selection of the best dairy cows was applauded by researchers and scholars 

(Alitaneh, Naeeimipour, & Golsheykhi, 2015). Though the AHP covers many of the 
existing issues in this area, it loses most of its functionality of correlation between 

various factors, since the assumption is not valid for studying the effects of the 

interior and exterior environment. This major limitation led the developer of the AHP 

method to work on and present the Analytic Network Process (ANP). This new 

method takes into account the intertwined relations of decision making elements by 

replacing the hierarchical structure with a network one. The Analytic Network 

Process is considered a more expanded version of the AHP (Saaty, 1999d). In 

general, in the AHP method the question that is asked in the pairwise comparisons is 

which of the 2 elements is more effective. For the same purpose, Saaty proposed that 

the intensity detected for inter-factor comparisons be graded on a scale of 9. Thus, 

based on relative significance the scale of 9 shows that a factor is more significant 

than another while the scale of 1 shows no difference or equal significance. In AHP, 

once the relative weight vectors are calculated, Saaty suggests a consistency rate (CR) 

of 0.1 for reliability and acceptance of a judgment on the pairwise comparisons 

matrix. Otherwise, further study of the problem and re-evaluation of the matrices is 

recommended.  

 

Briefly, Harker, and Vargas (1987) evaluated a quadratic and a root square. Lootsma 

(1989) argued that the geometric scale is preferable to the 1–9 linear scale. Salo and 

Hamalainen (1997) point out that the integers from 1-9 yield local weights, which are 

unevenly dispersed, so there is lack of sensitivity when comparing elements, which 

are preferentially close to each other. Based on this observation, they proposed a 

balanced scale where the local weights are evenly dispersed over the weight range 

[0.1, 0.9]. Earlier Ma, and Zheng (1991) calculated a scale where the inverse elements 

x of the scale 1/x are linear instead of the x in the Saaty scale (see Table 1). 

  



IJAHP Article: Alitaneh /Theories on coefficient of variation scales triangle and 

normalization of different variables: a new model in development of multiple criteria decision 

analysis 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

285 Vol. 11 Issue 2 2019 

ISSN 1936-6744 
https://doi.org/10.13033/ijahp.v11i2.565 

 

Table 1  

Different scales for comparing two alternatives 

 

Scale type Values 

Linear 

(Saaty,1977b) 
1 2 3 4 5 6 7 8 9 

Power 

(Harker, and 

Vargas, 1987a) 

1 4 9 16 25 36 49 64 81 

Root square 

(Harker, and 

Vargas,1987b) 

1 1.41 1.73 2 2.23 2.45 2.65 2.83 3 

Geometric 

(Lootsma,1989) 
1 2 4 8 16 32 64 128 256 

Inverse linear 

(Ma, and Zheng, 

1991) 

1 1.13 1.29 1.5 1.8 2.25 3 4.5 9 

Asymptotical 

(Dodd, and 

Donegan,1995) 

0 0.12 0.24 0.36 0.46 0.55 0.63 0.70 0.76 

Balanced 

(Salo and 

Hamalainen, 

1997) 

1 1.22 1.5 1.86 2.33 3 4 5.67 9 

Logarithmic 

(Ishizaka, 

Balkenborg, and 

Kaplan, 2006) 

1 1.58 2 2.32 2.58 2.81 3 3.17 3.32 

 

In this research and based on balanced applied theories of grading, some sort of a 9 

grade scale was proposed for scalar triangles of CV (CVST), and a type of new 

approach for de-scalarization of quantitative variables (normalization). The resulting 

triangular analysis has some special features, including: 

 

1. The ability to grade in higher scales based on the researcher’s decision. 

2. Reasonable and orderly mathematical structure (binomial expansion). 

3. Achieving a smaller consistency rate in comparison to Saaty’s CR. 

4. No limitation in terms of number of variables (n variables) through de-                    

   scalarization of measurement units based on a range of 1-9 scales.                       

5. Easy registration of analyzed data in software applications based on 1-9 scales. 

 

Generally, the relative scales triangles are based on CV of natural numbers, and since 

mean and SD parameters are taken together, the decision maker is able to choose the 

best criteria and target based on pairwise comparisons. Saaty (1977b) showed that the 

geometric mean is the most suitable mathematical rule for combining AHP 

judgements. In scale triangle theory however, a scale is used that takes not only the 

mean of natural numbers but also the variance and SD. Thus, it seems CV is a better 

and more precise method for normalization of numbers, followed by pairwise 

comparisons with the geometric mean or other methods. On the other hand, the AHP 

method uses nominal mean and scales (linear de-scalarization) for prioritization of the 



IJAHP Article: Alitaneh /Theories on coefficient of variation scales triangle and 

normalization of different variables: a new model in development of multiple criteria decision 

analysis 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

286 Vol. 11 Issue 2 2019 

ISSN 1936-6744 
https://doi.org/10.13033/ijahp.v11i2.565 

 

normalization process and pairwise comparisons. In the present paper, normalization 

and de-scalarization of quantitative variables is based on a scale triangle 

(multidimensional triangle). 

 

 

2. Data and methodology 

2.1 Coefficient of Variation (CV) 

When calculating the dispersion of data, one always faces data measured with various 

scales. Thus, to compare the dispersion of data collected from the statistical 

population measured with various scales, the use of only mean or variance does not 

provide accurate results as both are dependent on measurement scales. For the same 

reason, a more reliable scale such as coefficient of variation (CV) is used, which is 

calculated by dividing the standard deviation (SD) on the mean according to the 

following equation. 

 

                                       𝐶𝑉 =
𝜎

𝜇
                                                          (1) 

 

As can be observed, mean (µ) and standard deviation (σ) lack the required level of 
precision on their own, and may not be considered as the target criteria. The CV is 

free of dimensions and for the same reason it is suitable for comparison of statistical 

data with various measurement units. It is therefore quite good for normalization and 

de-scalarization of different variables and factors (Everitt, 1998). 

 

This paper attempts to explain the application of the scalar triangle in coordination 

and comparing various decisions. The scalar triangle helps decision makers 

conceptualize their decision options in an integrated manner using one scalar 

structural model, such that the optimal decision includes the ideas of all members and 

the decision maker. It may then be used as the best tool for making decisions in the 

shortest possible time. This paper provides an overview of the various stages of 

decision making and normalization of quantitative and qualitative variables using a 

scalar triangle in comparison to the hierarchical tree. 

 
2.2 Scales triangle 

The first step includes the creation of a triangular structure of natural numbers (n). In 

order to have a 9-step grading system the range of numbers was defined as 1-54 and 

the position of each number in the triangle was called a “cell” (0<n≤54). The purpose 

of this triangle is to express the studied problem in view of each topic and it needs to 

have easy software navigation in order to enable quick and optimum decision making. 

The triangle is graded in 9 lines. However, there is room for expanding the scaled 

spectrums and the structure of the scalar triangle theory based on n increased number 

of cells and lines (expansion of the definition of natural numbers in cells, lines and 

scales). Basically, each horizontal axis (x) of the triangle is a set of several numeral 

cells expressing one line of decision making (see Table 2). Thus, based on the 

triangular structure of the table below and the related descriptions, the triangular 

scales are calculated and analyzed in 9 lines.  

 

First, as a can be seen in the Table 2, for each line a mathematical function is defined 

based on the coefficients of binomial expansion, such that the computational potential 

of each line is an ordinal number specific to the same line. 

 



IJAHP Article: Alitaneh /Theories on coefficient of variation scales triangle and 

normalization of different variables: a new model in development of multiple criteria decision 

analysis 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

287 Vol. 11 Issue 2 2019 

ISSN 1936-6744 
https://doi.org/10.13033/ijahp.v11i2.565 

 

 

Table 2 

Natural numbers triangle 

 

Numbers Equation Line(l) 

 2 1          (𝑋𝑖 + 𝑌𝑗)
𝑙1  1 

 5 4 3  (𝑋𝑖 + 𝑌𝑗)
𝑙2  2 

 9 8 7 6  (𝑋𝑖 + 𝑌𝑗)
𝑙3  3 

 14 13 12 11 10  (𝑋𝑖 + 𝑌𝑗)
𝑙4  4 

 20 19 18 17 16 15  (𝑋𝑖 + 𝑌𝑗)
𝑙5  5 

 27 26 25 24 23 22 21  (𝑋𝑖 + 𝑌𝑗)
𝑙6  6 

 35 34 33 32 31 30 29 28  (𝑋𝑖 + 𝑌𝑗)
𝑙7  7 

44 43 42 41 40 39 38 37 36 (𝑋𝑖 + 𝑌𝑗)
𝑙8  8 

54 53 52 51 50 49 48 47 46  45 (𝑋𝑖 + 𝑌𝑗)
𝑙9  9 

 

 

Thus, if L and K are normal numbers in the presumed triangle, for (X+Y)
 L

 we have: 

 

L=1,2,…,n         K=1,2,…,m 
                                                                                             

(x+y)
L
=∑ (

𝐿
𝐾
)𝑋𝐿−𝐾𝑌𝐾𝐿𝐾=1  → (

𝐿
𝐾
) =

𝐿!

𝐾!(𝐿−𝐾)!
                          (2) 

 

The above equation is a fixed binomial coefficient. Hence, the setting of the binomial 

expansion triangle for each line was based on this. In the next step, the X and Y axes 

must be defined for each cell and the whole line numbers triangle. The basic 

assumption of the scalar triangle is that each cell is formed by the horizontal X vector 

(addition of all numbers on x axis) and vertical Y vector (multiplication of numbers 

on the Y axis). 

 

𝐴 = {(𝑥, 𝑦)|𝑥, 𝑦 ∈  {1,2, … , 𝑛}, 𝑥 = 𝑦}                                       (3) 
 

Moreover, to determine the exponent of each line, and if Nn is a set of natural 

numbers with a finite range, for each line we have the following condition: 

 

𝑁𝑛 = {1,2, … , 𝑛}          𝑓: 𝑁𝑛 → 𝐿             𝑓(𝑛) = 𝑙𝑛                 (4) 
 
Besides taking the 9 assumed lines in the triangle, the paper goes on to define the Ln 

condition of a set of natural numbers up to L9. Thus, the basics for the exponent of 

each line equation were defined according to Ln condition. Now, the final equation 

for each line of the triangle may be defined as follows: 

 

𝑇𝑖𝑗 = ∑ (𝑋𝑖 + 𝑌𝑗)
𝑙𝑛𝐿

𝑖=1                                                            (5)               
 

Now, whenever a total of X values and the numeric value of each cell is equal to the 

last line of the assumed triangle, we have, the X value on the horizontal axis of each 

cell of the triangle is equal to: 

 

𝑋𝑖 × 𝐶𝑖 = 𝐿9  → 𝑋𝑖 = 𝐿9 × 𝐶𝑖                                                      (6) 



IJAHP Article: Alitaneh /Theories on coefficient of variation scales triangle and 

normalization of different variables: a new model in development of multiple criteria decision 

analysis 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

288 Vol. 11 Issue 2 2019 

ISSN 1936-6744 
https://doi.org/10.13033/ijahp.v11i2.565 

 

 

On the other hand, if the result of multiplication of Y in the numerical value of each 

the cell is assumed to be equal to the last line of the triangle, we have, the Y value on 

vertical axis of each triangle cell is equal to: 

 

     𝑌𝑗 + 𝐶𝑖 = 𝐿9  → 𝑌𝑗 = 𝐿9 − 𝐶𝑖                                       (7)                                                                                           

 

To prevent dispersion, and in favor of the normal distribution of data and variables, a 

base 10 logarithm condition was added to the equation.  

 

𝑇𝑖𝑗 = log10[(𝐿9 × 𝐶𝑖 + 𝐿9 − 𝐶𝑖)
𝑙𝑛]                                       (8) 

 

Tij: the final number for each cell from numbers triangle equation, where each cell 

includes a presumed number (Tij). 

Cj : the assumed number for each cell in the scales triangle is between 1 and 54 

L9 : the assumed number for the last line of the triangle is the natural number 9 

ln : the exponent value assumed for each line, which is between natural numbers 1 and 
9 in the scales triangle.  

 

Finally, in order to make a unified and outstanding curve of the balance of the x and y 

vectors, each natural number existing in each cell (1 to 54) is calculated through the 

overall Ti equation, therefore creating a set of numbers with normal and balanced 

distribution (see Table 3).  

 

Table 3 

Balance numbers for the CVST 

 

(Xi + Yj)
2
 3.037 3.226 3.380 

       
(Xi + Yj)

3
 5.268 5.439 5.590 5.725 

      
(Xi + Yj)

4
 7.798 7.947 8.085 8.212 8.331 

     
(Xi + Yj)

5
 10.553 10.684 10.807 10.923 11.034 11.139 

    
(Xi + Yj)

6
 13.488 13.603 13.713 13.819 13.921 14.019 14.113 

   
(Xi + Yj)

7
 16.571 16.674 16.773 16.870 16.963 17.053 17.141 17.226 

  
(Xi + Yj)

8
 19.782 19.874 19.964 20.052 20.138 20.221 20.303 20.382 20.460 

 
(Xi + Yj)

9
 23.103 23.187 23.269 23.350 23.428 23.506 23.581 23.656 23.728 23.800 

 

Now, in one of the most significant steps and according to Equation (8), the total of 

the normal data for each is calculated for each line using the CV equation and the 

resulting numbers are introduced as random indices of the scales triangle. By finding 

a random index equation, one can attain the final goal of this paper, which is 

prioritization and special scaling/grading by means of a triangular structure of natural 

numbers (see Table 4). It must be noted that a random index is needed for the 

calculation of an inconsistency rate. Nevertheless, in the description and definition of 

this theory there is no need for the use of a random index in the scales triangle. 

Another advantage of this model is that it can make use of a random index of 

hierarchical analysis in the calculation of inconsistency rates of numbers triangle. 

 

  



IJAHP Article: Alitaneh /Theories on coefficient of variation scales triangle and 

normalization of different variables: a new model in development of multiple criteria decision 

analysis 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

289 Vol. 11 Issue 2 2019 

ISSN 1936-6744 
https://doi.org/10.13033/ijahp.v11i2.565 

 

Table 4 

Random index (RI) for the CVST 

 

 

 

In the final step, after the CV is calculated and the values of random indices are set, 

the final scale (grading) is also calculated in the scales triangle model through the 

following equation:  

 

Wij =
Ximax

Xij
                                                   (9) 

 
2.3 For example (line 2) 

𝑇𝑖𝑗 = log10[(9 × 3 + 9 − 3)
2] = 3.037 

𝑇𝑖𝑗 = log10[(9 × 4 + 9 − 4)
2] = 3.226  

𝑇𝑖𝑗 = log10[(9 × 5 + 9 − 5)
2] = 3.380  

𝐶𝑉𝑇 = 3.037 ,   3.226  ,    3.380 = 5.34  (RI) 

Wij =
9

5.34
= 1.6  (Equally to moderately) 

 

Where, Wij is the final scale or weight, Ximax is the largest relative index, and Xij is 

each of the relative indices. Therefore, the final scale was determined according to the 

analysis of a scales triangle of CV through formulation of natural numbers as 

described below (Table 5). 

 

   Table 5 

   Pairwise comparison scale for CVST preferences 

 

 

One of the most important advantages of using scale triangles is the lower 

inconsistency rate in comparison to AHP, which is an indication of its higher 

precision in the analysis of different variables. For instance, pairwise comparison A is 

related to AHP scaling and matrix B is related to scales triangle scaling (CVST).  

10 9 8 7 6 5 4 3 2 1 
Size of 

matrix 

0.87 1 1.15 1.35 1.62 2 2.61 3.58 5.34 9 RI 

Numerical rating (𝐖𝐢𝐣) Verbal judgments of preferences 

9 Extremely preferred 

7.8 Very strongly to extremely 

6.6 Very strongly preferred 

5.5 Strongly to very strongly 

4.5 Strongly preferred 

3.4 Moderately to strongly 

2.5 Moderately preferred 

1.6 Equally to moderately 

1 Equally preferred 



IJAHP Article: Alitaneh /Theories on coefficient of variation scales triangle and 

normalization of different variables: a new model in development of multiple criteria decision 

analysis 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

290 Vol. 11 Issue 2 2019 

ISSN 1936-6744 
https://doi.org/10.13033/ijahp.v11i2.565 

 

A= 

[
 
 
 
 

1 2 6 5 1/2
1/2 1 4 3 1/3
1/6 1/4 1 1 1/6
1/5 1/3 1 1 1/5
2 3 6 5 1 ]

 
 
 
 

     𝜆𝑚𝑎𝑥 = 5.083     CI=0.0208     CR=0.0186 < 0.1   

 

Dividing all the elements of the weighted sum matrices by their respective priority 

vector element, we obtain: 

  

(.2909, .1693, .0547, .0621, .4230) 

 

B= 

[
 
 
 
 
 
 1 1.6 5.5 4.5

1

1.6
1

1.6
1 3.4 2.5

1

2.5
1

5.5

1

3.4
1 1

1

5.5
1

4.5

1

2.5
1 1

1

4.5

1.6 2.5 5.5 4.5 1 ]
 
 
 
 
 
 

    𝜆𝑚𝑎𝑥 = 5.031   CI=0.0078   CR=0.0070 << 0.1 OK  

 

(.2953, .1805, .0615, .0708, .3919)  

 

Comparing vectors by compatibility: 

 

G= 94.02% > 90% Thus, priority vectors A and B are compatible vectors (equivalent 

vectors for measurement purposes). 

 

Moreover, the inconsistency rates calculated by the Saaty method and scales triangle 

method demonstrated that they had much better results than pairwise comparison A 

and scale triangle matrix B coefficient of variation. This proved the high precision 

and efficiency of the proposed method. In this method each cell is a function of an 

equation, and eventually the total of the equation of each given line is governed by 

the CV of the same line. 

 

Table 5 provides good insight into the comparisons and preferences (oral judgment) 

of a scales triangle containing the significance of various factors. In summary, the 

basis for a scales triangle may be described in a few stages:  

 

1. Creating a triangular structure of natural numbers. 

2. Creating 9 numeric lines for pairwise comparisons. 

3. Defining binomial expansion functions for each line of the triangle (X+Y)
 L

. 

4. Calculating or approximating numbers in each cell based on a logarithmic formula 

(formulation). 

5. Calculating the CV of a set of numbers produced by the formulation of each line 

(e.g. in line 1). 

6. Finding a random index for each line. 

7. Finding the final rates through the random indices equation. 

8. Calculating and achieving smaller inconsistency rates. 

9. Better efficiency and precision in analysis of qualitative variables. 

 
2.4 Normalization of variables 

One of the most important issues in MCDM is the existence of various scales for 

quantity and quality indices. Initially, it was just a lack of a standard for measurement 



IJAHP Article: Alitaneh /Theories on coefficient of variation scales triangle and 

normalization of different variables: a new model in development of multiple criteria decision 

analysis 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

291 Vol. 11 Issue 2 2019 

ISSN 1936-6744 
https://doi.org/10.13033/ijahp.v11i2.565 

 

of quality indices, for which Saaty used a linear de-scalarization technique. Instead of 

his technique, we have applied the scales triangle technique. The variable 

normalization method is also based on the scales triangle. In general, each variable 

must be normalized and correlated if it is to be measured and weighed along with 

other variables so that its measurement and weighting is in relation to all other 

variables. One the biggest problems of a decision maker is how to enter quantity 

values into related software (data bigger than 9 and less than 1).  

 

The “narrow” range of 1 to 9 should not be a problem when the model is correctly 

made as can be observed in research works related to medicine, biology and animal 

science. For instance, there are two dairy cows. Cow 1 has excellent body weight, a 

daily milk production of 110.27 lbs, a somatic cell count of 1459, and milk impurity 

of 0.83%. Cow 2 has medium body weight, a daily milk production of 95.08 lbs, a 

somatic cell count of 2610, and a milk impurity rate of 0.47%. The decision maker 

uses a software application to analyze quality variables based on existing scales, but 

is unable to make suitable analyses in a short time and with high precision in terms of 

milk production, somatic cells, and impurity rate. Also, the AHP/ANP software needs 

considerable time for pairwise comparisons between various groups due to the 

existence of various units and traits.  

 

In order to address such problems, the present study focuses on making precise and 

actual estimations of comparisons and attempts to develop a suitable model for 

normalization of quantity variables (de-scalarization). As noted earlier, when using 

CV for calculation of the mean there is a need for more than one data or variable, and 

since this is not possible for data available on dairy cattle (each cow has a record of 

various factors), one must look for a model capable of properly normalizing all 

records and variables based only on the one reported data. 

 

Let’s assume a set of rational numbers as a set whose numbers may be written in the 

following general format: 

    Q ={X= a/b | (a,b) Z , b < 0}                                           (10) 

In this method, the range of normalization of a quantity variable is a set of numbers 

bigger than zero (rational numbers) which must be standardized if the primary 

variable is to be normalized. Thus, the first practical step is to calculate the square 

root and then the inverse power of the last line of the triangle is used to formulate and 

finalize the two equations. It must be emphasized again that to analyze the CV of a 

number there must be more than one data, variable, record, parameter, etc. For the 

same reason, two equations were extracted from the scales triangle with the 

maximum continuity and correlation, and their CV was analyzed with minimum 

error.  Besides, in order to analyze two desirable and balanced equations Tq which is 

related to the mathematical model of scales triangle, the following steps were 

followed: 

 

 At first, if Q is taken as a number bigger than zero and its square root is 

calculated, we have:       X𝑞 = √Q 

 

 After that, using the equation of the last line of triangle and dividing the 
power of the first line by that of the last line, the first equation (Tq1) was 

produced, and then the second equation (Tq2) included the result of the first 

equation and the primary variable. At the end of this process, the two 



IJAHP Article: Alitaneh /Theories on coefficient of variation scales triangle and 

normalization of different variables: a new model in development of multiple criteria decision 

analysis 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

292 Vol. 11 Issue 2 2019 

ISSN 1936-6744 
https://doi.org/10.13033/ijahp.v11i2.565 

 

standard and balanced equations Tq1 and Tq2 were formulated with maximum 

correlation and reliability: 

 

        𝑇𝑞1 = log10[(𝐿9 × 𝑥𝑞 + 𝐿9 − 𝑥𝑞)
1/𝑙9

]                                       (11) 

 

        𝑇𝑞2 = log10[(𝐿9 × 𝑥𝑞 + 𝐿9 − 𝑇𝑞1)
1/𝑙9

]                                      (12) 

 
In these two equations, due to reduced significant error and maximum reliability 

of the analysis result, the values of Tq1 and Tq2 showed better numerical 

correlation, as confirmed by the correct and relevant results obtained from 

calculation of their numerical value and CV (2 variables) through Equation 13. 

Interestingly, in analysis of numbers bigger than 0 the CV was on a 9-grade 

scale. It must be noted that this mathematical model is capable of numerical 

calculations up to n numbers. It must be emphasized that the obtained CV was 

also normal and lacked statistical units (de-scalarization). 

 
𝐶𝑉𝑇 = ∑  𝑇𝑞1 

𝑛
𝑞>0 𝑇𝑞2                                                    (13) 

 

Generally, normalization by a scales triangle goes through the following stages:  

 

1. Getting the square root of the related data or variable. 

2. Formulating two equations based on the last line of the triangle. 

3. Finding more than one data (at least 2) for the CV calculation. 

4. Calculating the CV for the two data. 

5. Normalizing several variables with various scales, grading and performing 

pairwise comparisons. 

 

Based on this and a normalization procedure on scales triangle, for example the 

records of dairy cow production rates in Table 6, we have: 

 

Xq = √110.27 = 10.5 

𝑇𝑞1 = log10[(9 × 10.5 + 9 − 10.5)
1/9] = 1.0142 

𝑇𝑞1 = log10[(9 × 10.5 + 9 − 1.0142)
1/9] = 1.0564 

𝐶𝑉𝑇 = 1.0142  , 1.0564 = 2.88 (Normalized) 
 

Table 6 

Normalizing several variables with various scales 

 

Milk 

impurity (%) 

Somatic cell 

count(number) 

Daily milk 

(lb) 

Body weight 

(Quality trait) 
Cow 1 

0.83 1459 110.27 Excellent Data 

4.53 2.15 2.88 9 Normalized 

Milk 

impurity (%) 

Somatic cell 

count(number) 

Daily milk 

(lb) 

Body weight 

(Quality trait) 
Cow 2 

0.47 2610 95.08 Medium Data 

4.67 2.03 2.92 4.5 Normalized 

 

 



IJAHP Article: Alitaneh /Theories on coefficient of variation scales triangle and 

normalization of different variables: a new model in development of multiple criteria decision 

analysis 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

293 Vol. 11 Issue 2 2019 

ISSN 1936-6744 
https://doi.org/10.13033/ijahp.v11i2.565 

 

As shown above, in this stage all quantity variables are scaled by means of the scales 

triangle. Thus, the pairwise comparisons were performed with high speed and 

accuracy in comparison to other methods. 

 

 

3. Conclusion 

One condition for the efficiency of the above process is to prioritize research plans 

and projects using methods suitable to prioritization. Based on the present paper, the 

theory of scales triangle (CVST) and normalization of quantitative and qualitative 

variables shows a high potential for extensive use in various fields. This is due to its 

high capacity in modeling actual problems, high speed of analysis and the ease of 

learning for users. Though mathematical and statistical methods provide optimized 

results for planning and decision making, such techniques and models often require 

precise and definite data, which is a serious issue in actual conditions where it is 

difficult to collect data and takes a lot of time. Hence, it may be said with near 

certainty that this new proposed method may play a major role in grading, 

normalizing (descalarization) of different variables and making hard decisions. It 

seems the theories of scales triangle and normalization of numerical variables is an 

integrated model for analysis for quantitative and qualitative variables. Particularly, 

these theories are designed to show newer, better and faster paths with high usability 

potential and ease of software navigation. Since grading methods of other scholars are 

based on Saaty’s 1-9 scale analysis, it is safe to claim that the new model is 

independent of any existing grading methods. The scales may be increased or 

decreased based on increased number of cells and lines in the scales triangle (noting 

that while solving problems the equations must be calculated according to the last line 

(ln) of the triangle). The present paper attempted to ease the use of related software 
by prioritization into a 1-9 scale. Efforts were also focused on putting the scaling into 

a process that is dependent on data and final weights.  

 

In the end and given the wide use of AHP/ANP, it seems that more future works will 

focus on the scales triangle method. It is imperative to note that such decision making 

techniques, like all other methods, only serve to convert data into information for 

decision makers, and it is the decision maker who has to make the best choice. The 

method proposed here may lead to better results and changes in pairwise 

comparisons. The paper demonstrated, on the other hand, that the structure of a 

numbers triangle is capable of including and analyzing several equations at once. This 

may lead more researchers into the field in the near future. The author hopes these 

results play a significant role in the expansion and clarification of comparisons and 

decisions, and finally creates a unified framework for proper selection and a new 

applied method for multiple-criteria decision making analysis. 

 

Finally, the results of this research in relation to a new scale, known as the coefficient 

of variation scales triangle (CVST), suggest it can be used along with other scales. 

Also, the CVST model can be identified as a suitable and practical method. 

 
 

 

 

 

 

  



IJAHP Article: Alitaneh /Theories on coefficient of variation scales triangle and 

normalization of different variables: a new model in development of multiple criteria decision 

analysis 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

294 Vol. 11 Issue 2 2019 

ISSN 1936-6744 
https://doi.org/10.13033/ijahp.v11i2.565 

 

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